c Heldermann Verlag ISSN 0940-5151
Economic Quality Control Vol 16 (2001), No. 2, 255 – 269
Optimal Productivity and Investments in Quality: An Operations Parametric Model Shailesh S. Kulkarni, Maliyakal D. Jayakumar and Victor R. Prybutok
Abstract: In this paper we present a mathematical model of a generic manufacturing system. The quality of the manufactured product is captured through one or more of its critical design/ process parameters. Managers often face the dilemma of which product/process is to be singled out for improvement in quality, with the limited capital outlay on hand. In this paper, we use the critical process parameters along with the standard production variables in a mathematical programming framework, to identify the process to be targeted for improvement. Consistent high quality, results in higher rewards in a perfectly competitive market place and also requires higher amounts of the employed resources, including capital. The optimal investment to be made in achieving higher rewards depends upon the product characteristics. We consider alternative systems, which differ in their costs of quality, since the underlying process parameters follow different probability distributions. Our experiments provide some important insight into the optimal investments in quality, and the accompanying qualityproductivity trade-offs. We demonstrate that 100% process conformance or 100% use of the productive resources do not result in maximum net profits. Our model also reinforces the notion that consistent high quality ultimately translates into a corresponding gain in productivity and higher profits or net revenues. Key Words: Quality, Productivity, Process Investment, Quality Improvement, Non-Linear Optimization.
1
Introduction
In the past three decades, enterprises worldwide have taken major strides in integrating Quality and Productivity into their operational economics. Researchers in this area have long recognized the complexities of integrating these two critical elements (Cooke 1989, Merino, 1990), in any viable mathematical model. Considering that each of these elements, viz., Quality and Productivity presents a multifaceted attribute, it is not surprising that the literature examining their quantitative interrelationships continues to be sparse. Discussion of the theory underlying the integration, case methods, and some anecdotal issues are cited in Maani, 1989. A basic Geometric Programming model for the process was proposed earlier by Jayakumar, Prybutok, and Guzek, (1993). After establishing the framework, the authors suggested that the different functional relationships between quality costs/profits could be
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explored with the model. In this paper, we extend the earlier model to explore some such functional relations. We assign different probability distributions that could be attained, for different processes, and link these to corresponding capital investments. The results from the model indicate the optimal investments, and the attainable quality that would maximize their returns (net profits). The foundations of our work wrest in common sense. The original contribution of this work is the modeling methodology, comprising 1. the use of clearly identifiable and critical product traits to capture the product’s quality characteristics in summary form, 2. tying quality parameters with other process variables in a non-trivial model, 3. the geometric program that explicitly addresses the alternative investments in quality, in their functional form, and, 4. the insights gained from the resulting parametric models, some of which appear to be quite counter-intuitive. We believe this work provides an important step in understanding the integration of quality and productivity into the operations of any profit maximizing enterprise. Improvements in quality and/or productivity will be pursued only to the extent dictated by long-term growth in revenues and net profits. In the next section we present a review of existent research. In Section 3, we develop a simple mathematical model of a production system. For this model, Quality is associated with stricter conformance to the expected, rather than the best possible performance. The model addresses the optimal levels of quality and productivity functions, in the framework of maximizing the net profits from the firm’s operations. Our model explicitly recognizes that the investments to be made in Quality should result directly in improving the critical product traits, by reducing their variability. We then provide numerical examples assigning different probability distributions to these underlying quality traits in Section 4. The examples illustrate the concepts and produce some new insights. Section 5 concludes the paper with a summary.
2
Background
It is important to define quality of any product in terms that are consistent with the processes that back its production, and the processes where it is used. The five general definitions of quality discussed by Garvin (1984) are: • transcendent: innate excellence • product-based: relates quality improvement to increases in the quantity of a product attribute
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• user-based: fitness of the product for intended use • manufacturing-based: conformance to the product specifications • value-based: the compromise between the actual cost and the perceived value (in the use) of the product Accordingly, we consider some characteristic traits of the product to model its quality. In general, the quality traits used may be so diverse as the titanium content of the material used to make a part, or its finished size, or some such, depending on its end use. Consistent with the existing literature, quality is operationalized as the percentage of conforming items as set by the specifications of these traits. To keep the model simple, we start the model building with each product having only one such characteristic trait described by the operational parameters. For example: in case of a ball bearing assembly the critical product parameter may be its inner diameter. We view quality as a continuous variable that results in a proportion of the output being acceptable at each stage of the production/service process. Similar to the specifications for products, inputs such as raw materials also may be subject to specific quality criteria. A better quality process should result in lower variability of the selected operational parameter, at every stage. The investment costs for such a system should be expected to be higher (relative to the ones that permit larger variations in the product quality parameters). Mehrez and Myers (1994) used linear regression to estimate minimal costs of inputs required to attain a preset lower bound of the confidence interval of the output-volume, working within a fixed budget. Metzger (1993) used Data Envelopment Analysis (DEA) methodology to develop a model to measure the relative effectiveness of marginal spending, in which productivity costs and quality costs were two explicit components. Traditionally, productivity of a system is defined as the ratio of total standard hours earned to the total hours available from the system. The total standard hours earned is derived in two steps: 1. multiply the total number of units produced (that meets the customer quality requirements) by the corresponding time standard, and, 2. sum the result at step 1, across all output products. In the optimization context, this appears to be quite a circular approach. Total hours available include total paid hours less vacation, sick leave, authorized leave, etc. There are many caveats in the determination of the acceptable time standards for different products. Even after discounting these, maximization of productivity established in this manner may result only in the improvement of some measure of system utilization, and any maximization of this parameter need not result in the most profitable operation of the manufacturing system.
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In this work, productivity is identified simply with the volume of production. Our work identifies productivity in its economic sense, as an output to input relationship, and is a natural characteristic of the operating state of any production system. We contend that any firm directs its operations toward maximizing net profits, while working within the constraints imposed by available resources. Among inputs, we include all the resources that are required to operate the product lines. At every step in the production process, the product either meets or fails to meet the specifications. The probability of the product meeting the specifications, in turn depends on the quality and quantity of resources expended on the product. The end result of the process is either an acceptable product/service that meets customer quality requirements; or an unacceptable product/service that results in customer dissatisfaction and an associated loss of future sales. For the model developed here, productivity is considered only implicitly, as only a derived measure of resource utilization. The customer’s expectation of the quality requirement of any product depends on the cost incurred by him. To illustrate, contrast two situations that are consistent with our assumptions. In the first situation, assume the customer pursues the lower cost product; he may usually be purchasing an item with a higher probability of substandard performance (higher variance of the underlying quality parameter, with a rare chance of superior performance). For the second situation, consider a customer willing to incur a higher cost for the same product; his expected reward for the additional cost is a product that has a commensurately higher probability of superior performance (with a very rare chance of substandard performance, with a lower variance of the underlying quality parameter). For example, the Craftsman branded hardware marketed by Sears Roebuck commands a 200 to 500% premium over similar non-branded hardware though they mostly serve similar applications. The expected life (an exponential process) of the hardware in use (Craftsman often comes with lifetime replacement warranty) may be the criterion used to represent quality. For the models presented in this paper, the percentage non-conforming to the specifications is considered as a surrogate for the perceived quality. It may be argued that the proportion of defectives represents the consistency of the production process. Eventually, a process that has larger process variances and hence larger percentage defectives will result in higher overall costs for even the non-defective items. Recently, Ganeshan, Kulkarni and Boone (2001) and Kulkarni and Prybutok (2001) developed models for investing the process in order to improve its quality and reduce process variance. This view of stricter quality conformance at a higher cost is also consistent with Garvin’s (1984) value-based definition, cited above. Current views on the subject hold relative perceived quality as the single most important factor in determining the long term competence of any product, and thereby the manufacturing system. We maintain that products with consistently higher quality should command higher prices in the market place, and vice versa. This implies that quality levels should be set to meet the customer’s relative expectations while concurrently maximizing the producer’s profits. This approach should not require zero defects. In situations where zero defects are expected, the customer should generally be willing to pay higher (at times, inordinately or disproportionately
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high) prices. (This arises perhaps due to other practical concerns such as ability to perform under critical conditions, safety etc.) In this sense, the manufacturer should be concerned with reducing the variance of the process, thereby reducing the percentage non-conforming in the product lines, than indiscriminately reducing their number (or equivalently, percentage or fraction). This is consistent with the ideas of Deming (1986) who stresses prevention over detection of quality problems and exhorts ”building in high quality, not inspecting out poor quality.” He states that quality is best controlled at the process level and that quality is best improved by studying and changing production processes. (For the model developed here, we assume the underlying customer expectations of quality as functions of costs, for individual products.) However, the key to integrating productivity and quality, is to recognize that the belief that higher emphasis on quality always results in higher productivity (Deming, 1986) is not necessarily true. We suggest that beyond some optimal level, attempts to improve quality may actually result in lower productivity/ profitability. This position appears to differ from the philosophies of well-known quality experts such as W. Edwards Deming (1986) and Philip B. Crosby (1979) because they advocate that any improvement in quality will always lead to an increase in productivity. Simply stated, perfect (100%) quality essentially requires infinite cost. The perils of setting excessive quality standards are discussed in Juran (1999). These emerge on restricting the consideration to the highest quality processes, (Prybutok, Atkinson, and Saniga, 1990) where the investigation pertains to a narrow range of quality (where defects approach zero). The model developed in this paper is simple, but provides some new and interesting insights.
3
Model Development
The manufacturing/production facility has n distinct products. Each product consumes a fraction of the m input resources. These resources may be raw materials, labor, machines, financial resources, etc. All the tangible resources that are part of these processes are included in this group of size m. Initially, let us assume that the quality of each input resource is stationary at a specified level. This is done only to simplify the model. The assumption is valid for resources such as capital and labor because such items are generally not available at different quality levels. Subsequently, we will generalize this simple model to accommodate the variability associated with the quality level of some/all of the inputs such as raw materials. For the benefit of the reader we start with the basic model as proposed by Jayakumar, Prybutok, and Guzek (1993). Let us assume that the availability of the ith resource is restricted by an upper bound, RUi . Let DUj represent the upper bound on the demand of the jth product. Given the general definition of the problem, this demand will depend on the quality level of the respective product. Let dj be the fraction defective (i.e., nonconforming) in the product line, of jth item, so that the percentage of defect free items (Qj ) in the output can be represented by (1 − dj ). This definition is consistent with the traditional approaches.
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Let xj denote the level of production of the jth product. Let aij be the requirement of the ith resource for the production of one unit of defect-free jth product. (This requirement corresponds to the quality level, Qj = 1, the maximum that is possible.) It is imperative that the optimal value of xj should depend on the level of quality corresponding to the output. Let us denote the true unit consumption of the ith resource for the jth product as rij for any pair i, j. It should be possible to represent rij as a function of the product’s quality. As a first step, we develop a simple model by assuming that this relationship is of the multiplicative form such as: r
aij Qj ij xj
(1)
The exponent rij introduced above may be considered as the consumption exponent and its values should be greater than 1 for most inputs (such as machine time, labor etc.), reflecting that their consumption should increase more than proportionately with increasing levels of quality. For the basic raw material inputs, the consumption exponent may be equal to one. The above relationship is particularly good for modeling the increasing rate of consumption of resources as the quality levels approach 1 (100% defect free). This simple relationship is reasonably representative of reality, and affords mathematical simplicity. Within this framework, the demand and resource restrictions translate into the following constraints. n n r aij Qj ij xj ≤ RUi or aij (1 − dj )rij xj ≤ RUi for any i (2) j=1
Qj xj ≤ DUj
j=1
or (1 − dj )xj ≤ DUj
for any j
(3)
The first set of constraints states that the resources consumed have to be within available limits. The second set of constraints states that the production has to be within the targets specified. The nature of the problem also requires that all xj ’s be nonnegative. Clearly, dj will be an element of the unit-interval by definition. We assume that the objective of the firm is to maximize profits, and that this corresponds to maximization of the total net revenues (i.e., the revenues associated with the products less all assignable costs), while working within the constraints imposed by the available resources. This represents a reasonable approach in situations where the unit profit margins on different products are not exactly known. However, when profit margins are known, they may be used to improve the validity of the model. The conventional approach of maximizing the productivity level or equivalently, the facility utilization level, may result in sub-optimization with respect to either one of these objectives. To model the objective function, we represent this maximum gainable unit net revenues (corresponding to the ideal defect free product) per unit item of the jth product by Cj . The actual unit revenue, should be subject to variation depending on the quality of the product and in general, should be lower than Cj . Let us represent this variation in a functional form as: Cj (1 − dj )pj
for any j
(4)
We refer to the positive exponent pj as the return exponent. This is similar to the consumption exponent defined in the case of constraint coefficients. In this case, pj > 1
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will indicate the willingness of the customers to pay more than proportionately for a consistent higher quality, and pj < 1 will represent situations where customers penalize the product less than proportionately for any deterioration in quality. The objective function of the algebraic formulation is to maximize the total revenues through the quantities produced in this manner. We recognize that there will be added cost for keeping quality at any selected level. For extending the basic model to include this added, process-dependent cost we proceed as follows: Let, yj be the key quality characteristic (for example: the inner diameter of a metal cylinder, or more generally the useful lifetime) of the product. Let σ 2j be the variance in the key quality characteristic of the conforming items of the jth product. Accordingly, the proportion of defectives, dj , can be identified as: LSL ∞ dj = fj (yj |σ j )dyj + fj (yj |σ j )dyj −∞
(5)
U SL
where, LSL and U SL represent the lower and upper specification limits, for the jth product, and fj (·|σ j ) is the parametric (the variance parameter) probability density function of the quality characteristic yj . We postulate that σ 2j should be a function of the investment that we are willing to undertake in attaining quality consistency in the jth product. It is natural to assume that this should be an inverse relation, i.e., the costs should be higher for attaining a smaller variance (or standard deviation). Let the upper bound on the total investment budget be, I. Then we have: n Ij ≤ I and σ 2j = gj (Ij ) (6) j=1
where, Ij is the investment in the jth product quality characteristic and gj (Ij ) represents the functional relation of σ 2j to Ij . One possible representation (and which we assume in our numerical experiments), could be σ 2j = σ 2Lj + (σ 2Mj − σ 2Lj )e−bj Ij
(7)
where, bj the jth exponent (this may be interpreted as the exponent of return on the investment in quality; the higher its value, the lower the effect in improving the quality for any fixed investment), is positive, and translates Ij to a corresponding σ 2j . This functional form allows for the attainment of the lower bound variance of σ 2Lj when Ij is infinite, or deteriorates to the maximum variance of σ 2Mj at Ij = 0. Also, it can be easily demonstrated that for this type of functional form, the proportion defectives (dj ) is monotonically decreasing in the investment. We can now re-assemble the earlier formulation as: n n max Cj (1 − dj )pj xj − Ij (8) xj ≥0,Ij ≥0,j∈{1,...,n}
j=1
subject to n aij (1 − dj )rij xj ≤ Ri
j=1
for all i = 1, . . . , m
(9)
j=1
(1 − dj )xj ≤ Dj
for all j = 1, . . . , n
(10)
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n
Ij ≤ I
(11)
j=1
This general model presents the minimization of a non-concave function over a nonconvex domain. Mathematically, the optimum of this problem does not necessarily occur at boundary points, and is hard to determine. Fortunately, we have found this objective function to be rather well behaved over a large range of parameter values and although we cannot conclusively prove global optimality, a few illustrative examples can generate some good insights. These are presented in the next section.
4
Numerical Experiments
In order to provide a numerical illustration of the model, we consider a hypothetical facility that has two products with one shared resource. The maximum unit resource requirements aij are [3, 2], the consumption exponents rij are [3, 3], the return exponents pj are [1, 1], the demand vector Dj is [30, 40], the profit coefficients Cj of the objective function are [3, 5], and the upper bound on the available resource and maximum possible investment are 400 and 60 units respectively. The conformance specification for the quality characteristic of product 1 is: LSL = 102, U SL = 112. The conformance specification for the quality characteristic of product 2 is: LSL = 100, U SL = 105. The characteristics of the production processes for both products are normally distributed with means µ1 = 105 and µ2 = 103, respectively. The upper and lower bounds on the variance are: σ 2Mj = 3 and σ 2Lj = 0.1 for j = 1, 2. The exponent for product 2, i.e., b2 was maintained at 0.08 and the exponent for product 1, i.e., b1 , was varied from 0.08 to 0.19 in increments of 0.01. The corresponding optimal production levels, profit and investments are presented in Table 1. The model was implemented in a spreadsheet and a standard solver (GRG) was employed to find the optimal solutions. Thus, the set of input parameters and input conditions is summarized in the following table: j a1j 1 3 2 2 j LSL 1 102 2 100
r1,j 3 3 U SL 112 105
pj Dj Cj 1 30 3 1 40 5 model bj 2 N (105, σ 1 ) 0.08(0.01)0.19 N (103, σ 22 ) 0.08
with the additional conditions: RUi ≤ 400 I ≤ 60 0.1 = σ 2Lj ≤ σ 2j ≤ σ 2Mj = 3
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From Table 1, it may be observed 1. that as the exponent of product 1 increases, the profits increase and asymptotically reach an upper limit, even while there are slack funds available for investing in quality (Only less than 30 units of the possible 60 units have been invested.) This indicates that the firm may find it profitable to invest in the processes only up to a certain critical level even though budget limitations are still slack. In short, the quality improvement, which yields optimal profits, need not be one, which utilizes the maximum investment. (Mathematically this could probably be explained by parametric analysis of the objective function.) 2. The Table shows that as b1 increases (i.e., as there is higher return to be expected for investing in the quality of product 1, greater ”bang for the buck”) the optimal unit production (x1 ) gets reset to a lower level. There is an increased investment in improving the quality of product 1 (higher I1 ), but this translates to a lower level of production (x1 ). 3. The change in b1 does not appear to have any influence on x2 .
Table 1 Process levels, investments and profits for the normal distribution. b1 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
x1 32.229 31.801 31.504 31.285 31.119 30.988 30.883 30.796 30.724 30.663 30.610 30.565
x2 40.836 40.836 40.836 40.836 40.836 40.836 40.836 40.836 40.836 40.836 40.836 40.836
I1 2.166 2.986 3.447 3.704 3.837 3.894 3.902 3.881 3.836 3.781 3.717 3.648
I2 Total Return 25.667 251.849 25.664 252.156 25.665 252.497 25.665 252.839 25.665 253.168 25.665 253.477 25.665 253.766 25.665 254.035 25.665 254.284 25.665 254.515 25.665 254.729 25.665 254.928
A parametric plot of the ratio of investments II12 into process 1 and 2 against the productivity exponent b1 , holding the exponent b2 at 0.08, is shown in Figure 1, which again illustrates that there is an optimal investment vector for the product mix, depending only on the relative merits of the investments.
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Figure1: The investment ratio as function of the productivity exponent for product 1 based on a normal distribution. In summary, for normally distributed processes, incremental variance reduction appears to result in exponentially lower returns. One question that could be raised at this stage is regarding the behavior of the ratio II12 . In specific, is the behavior in Figure 1 and in Table 1 specific to a Gaussian (Normal) process distribution? In order to answer this question, at least in part, we conducted some more experiments. First, assuming that the production processes for both products are uniformly distributed and then assuming that they follow a Gamma distribution. The results for the uniform distribution are presented below in Table 2 and with respect to the ratio II12 in Figure 2.
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Table 2: Process levels, investments and profits for the uniform distribution. b1 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
x1 41.715 39.009 37.495 36.499 35.785 35.246 34.822 34.480 34.197 33.960 33.758 33.583
x2 44.828 44.827 44.827 44.827 44.827 44.827 44.827 44.827 44.827 44.827 44.827 44.827
I1 19.057 20.858 21.170 20.914 20.418 19.821 19.192 18.562 17.949 17.362 16.803 16.273
I2 Total Profit 36.415 187.715 36.415 190.404 36.415 192.888 36.415 195.108 36.415 197.079 36.415 198.831 36.415 200.392 36.415 201.791 36.415 203.051 36.415 204.190 36.415 205.226 36.415 206.171
Figure 2: The investment ratio as function of the productivity exponent for product 1 based on a uniform distribution.
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The results for the Gamma distribution are presented below in Table 3 and with respect to the ratio II12 in Figure 3. Table 3: Process levels, investments and profits for the Gamma distribution. b1 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19
x1 37.692 35.640 34.503 33.755 33.218 32.812 32.494 32.238 32.027 31.850 31.700 31.571
x2 40.899 40.899 40.900 40.899 40.899 40.899 40.899 40.899 40.899 40.899 40.899 40.899
I1 24.701 26.373 26.561 26.182 25.556 24.823 24.051 23.277 22.519 21.789 21.090 20.425
I2 Total Cost 31.019 211.516 31.021 213.968 31.017 216.277 31.020 218.390 31.020 220.309 31.020 222.047 31.020 223.625 31.020 225.060 31.020 226.369 31.020 227.566 31.020 228.666 31.020 229.679
Figure 3: The investment ratio as function of the productivity exponent for product 1 based on a Gamma distribution. In answering the question raised earlier, we found that the pattern of the solution does not depend on the functional form assumed for the process distribution. Specifically, even with uniformly or gamma distributed processes, incremental variance reduction results
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in exponentially lower returns. This outcome is counter-intuitive, and could be very important in making decisions on capital investments in quality. Based on our experiments and Figures 1, 2 and 3 and Tables 1, 2 and 3 we have the following important observations: 1. We used Gaussian, Uniform and Gamma distributions to model the critical process. However, irrespective of the distributional nature of the underlying process, any incremental variance reduction always resulted in exponentially lower returns. 2. The best (or profit maximizing) choices may not occur at levels where, a. All the demand is satisfied. b. All resources are fully utilized (implying high productivity with respect to resource utilization). c. All the budgeted finances are invested in variance reduction (quality consistency) measures. At this point it is also important to note that the parameter “b” may incorporate implicit process improvement efforts such as TQM and Kaizen. The Tables and Figures presented above are also indicative of the fact that such efforts will result in an increase in profitability while requiring relatively less monetary investment in the process.
5
Summary
This paper presents an optimization model for evaluating alternative investments in quality. The original contribution of this work comes from: 1. the use of some clearly identifiable and critical product traits to capture its quality characteristics in summary form, 2. tying quality parameters with other process variables in a non-trivial model and, 3. the insights gained from the resulting parametric models. In addition, the model has some special features which merit special mention. (1) Although we have used one form of variance reduction function in order to model the investment in quality consistency, the model is general enough to accommodate any other type of variance reduction function. (2) The model is general enough to incorporate more than one quality characteristic (though complexity could be an issue).
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(3) The more general issue of stochastic demand and stochastic cost/revenue parameters for the products can be accommodated with some minor modifications to our model. We have conducted some experiments using stochastic parameters and recognize that it considerably complicates the analysis. We do not present the results in this paper for the sake of brevity of exposition. In conclusion, there are several important issues, which can be investigated, and we intend to continue work along the extensions outlined as part of our future research.
Acknowledgement: The authors would like to thank the two anonymous reviewers whose comments have significantly improved this paper.
References [1] Cooke, W. N. (1989): Improving Productivity and Quality Through Collaboration. Industrial Relations, 28, 299-319. [2] Crosby, P. B. (1979): Quality is Free: The Art of Making Quality Certain. McGrawHill, New York. [3] Deming, W. E. (1986): Out of the Crisis. MIT Press, Cambridge, MA. [4] Ganeshan, R., Kulkarni, S.S., Boone, T. (2001): Production Economics and Process Quality: A Taguchi Perspective. International Journal of Production Economics, 71, 343-350. [5] Garvin, D. A. (1984): What Does Product Quality Really Mean? Sloan Management Review, 26, 25-43. [6] Jayakumar, M.D., Prybutok, V.R., and Guzek, R.D. (1993): Integrating Quality and Productivity to Improve Net Profits: A Management Science Approach. Economic Quality Control, 8, 82-95. [7] Juran, J.M., Godfrey B.A., (Ed.) (1999): Juran’s Quality Control Handbook, 5th Edition, McGraw Hill, USA. [8] Kulkarni, S.S, Prybutok, V. (2001): Process Investment and Loss Functions: Models and Insights. Working Paper, University of North Texas. [9] Maani, K.E. (1989): Productivity and Profitability Through Quality - Myth and Reality. International Journal of Quality and Reliability Management, 6, 11-23. [10] Mehrez, A., Myers, B.L. (1994): A Note On Input Control Model of a Predictive Regression. Journal of the Operational Research Society, 45, 354-357.
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[11] Merino, D. M. (1990): Economics of Quality: Choosing Among Prevention Alternatives. International Journal of Quality and Reliability Management, 7, 13-26. [12] Metzger, L.M. (1993): Measuring Quality Cost Effects On Productivity Using Data Envelopment Analysis. Journal of Applied Business Research, 9, 69-79. [13] Prybutok, V, Atkinson, M, and Saniga, E. M. (1990): Sampling Strategies for Competing Quality Objectives. Production and Inventory Management Journal, 31, 49-53.
Shailesh S. Kulkarni (
[email protected]) Maliyakal D. Jayakumar (
[email protected]) Victor R. Prybutok (
[email protected]) Business Computer Information Systems Department University of North Texas P.O. Box. 305249 Denton, TX 76203-5249